Percolation of hard disks

Random arrangements of points in the plane, interacting only through a simple hard core exclusion, are considered. An intensity parameter controls the average density of arrangements, in analogy with the Poisson point process. It is proved that at high intensity, an infinite connected cluster of excluded volume appears with positive probability.Comment: 16 pages, 6 figure

First Order Phase Transition of a Long Polymer Chain

We consider a model consisting of a self-avoiding polygon occupying a variable density of the sites of a square lattice. A fixed energy is associated with each $90^\circ$-bend of the polygon. We use a grand canonical ensemble, introducing parameters $\mu$ and $\beta$ to control average density and average (total) energy of the polygon, and show by Monte Carlo simulation that the model has a first order, nematic phase transition across a curve in the $\beta$-$\mu$ plane.Comment: 11 pages, 7 figure

Dilatancy transition in a granular model

We introduce a model of granular matter and use a stress ensemble to analyze shearing. Monte Carlo simulation shows the model to exhibit a second order phase transition, associated with the onset of dilatancy.Comment: Future versions can be obtained from: http://www.ma.utexas.edu/users/radin/papers/shear2.pd

Weighted ensemble: Recent mathematical developments

The weighted ensemble (WE) method, an enhanced sampling approach based on periodically replicating and pruning trajectories in a set of parallel simulations, has grown increasingly popular for computational biochemistry problems, due in part to improved hardware and the availability of modern software. Algorithmic and analytical improvements have also played an important role, and progress has accelerated in recent years. Here, we discuss and elaborate on the WE method from a mathematical perspective, highlighting recent results which have begun to yield greater computational efficiency. Notable among these innovations are variance reduction approaches that optimize trajectory management for systems of arbitrary dimensionality.Comment: 12 pages, 10 figure

A cluster expansion approach to exponential random graph models

The exponential family of random graphs is among the most widely-studied network models. We show that any exponential random graph model may alternatively be viewed as a lattice gas model with a finite Banach space norm. The system may then be treated by cluster expansion methods from statistical mechanics. In particular, we derive a convergent power series expansion for the limiting free energy in the case of small parameters. Since the free energy is the generating function for the expectations of other random variables, this characterizes the structure and behavior of the limiting network in this parameter region.Comment: 15 pages, 1 figur

Statistical mechanics of glass transition in lattice molecule models

Lattice molecule models are proposed in order to study statistical mechanics of glass transition in finite dimensions. Molecules in the models are represented by hard Wang tiles and their density is controlled by a chemical potential. An infinite series of irregular ground states are constructed theoretically. By defining a glass order parameter as a collection of the overlap with each ground state, a thermodynamic transition to a glass phase is found in a stratified Wang tiles model on a cubic lattice.Comment: 18 pages, 8 figure

Steady-state kinetic studies with the polysulfonate U-9843, an HIV reverse transcriptase inhibitor

The tetramer of ethylenesulfonic acid (U-9843) is a potent inhibitor of HIV-1 RT * and possesses excellent antiviral activity at nontoxic doses in HIV-1 infected lymphocytes grown in tissue culture. Kinetic studies of the HIV-1 RT-catalyzed RNA-directed DNA polymerase activity were carried out in order to determine if the inhibitor interacts with the template: primer or the deoxyribonucleotide triphosphate (dNTP) binding sites of the polymerase. Michaelis-Menten kinetics, which are based on the establishment of a rapid equilibrium between the enzyme and its substrates, proved inadequate for the analysis of the experimental data. The data were thus analyzed using steady-state Briggs-Haldane kinetics assuming that the template:primer binds to the enzyme first, followed by the binding of the dNTP and that the polymerase is a processive enzyme. Based on these assumptions, a velocity equation was derived which allows the calculation of all the specific forward and backward rate constants for the reactions occurring between the enzyme, its substrates and the inhibitor. The calculated rate constants are in agreement with this model and the results indicated that U-9843 acts as a noncompetitive inhibitor with respect to both the template:primer and dNTP binding sites. Hence, U-9843 exhibits the same binding affinity for the free enzyme as for the enzyme-substrate complexes and must inhibit the RT polymerase by interacting with a site distinct from the substrate binding sites. Thus, U-9843 appears to impair an event occurring after the formation of the enzyme-substrate complexes, which involves either an event leading up to the formation of the phosphoester bond, the formation of the ester bond itself or translocation of the enzyme relative to its template:primer following the formation of the ester bond.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42867/1/18_2005_Article_BF01992044.pd